3-(3{40 ,5{40 -dihalogenophenyl)imidazolidine-2,4-dione derivatives

ABSTRACT

3-(3&#39;&#39;,5&#39;&#39;-Dihalogenophenyl)imidazolidine-2,4-dione derivatives, which may be substituted with alkyl or phenyl in the 1 and/or 5 positions, are new compounds, and have high microbicidal activity on various fungi and bacteria and are non-toxic to plants and mammals. Those compounds are prepared by the intramolecular cyclization of a corresponding urea derivatives.

United States Patent 1 June 6,1972

Fujinami et al.

inventors: Akira Fujinami, Ashiya-shi; Toshiaki Assignee: SumitomoChemical Company, Ltd.,

Osaka, Japan Filed: Nov; 14, 1969 Appl. No.: 876,987

Foreign Application Priority Data Nov. 25, 1968 Japan ..43/86483 May 13,1969 Japan ..44/37l60 U.S. Cl. ..260/309.5, 260/465 D, 260/470, 260/471R, 260/516, 260/518 R, 424/273 Int. Cl. ..C07d 47/32 Ozaki,Toyonaka-shi; Fukashl Horluchi, Saitarna-ken; Slgeo Yamamoto,Toyonakashi; Keiiehiro Aklba, lkeda-shi; Katsutoshi Tanaka,Takarazuka-shi; Shlgehiro Ooba, Takarazuka-shi; Tadashl Oolshi,Minooshi; Katsuji Nodera, Nishinomiya-shi; Nobuyuki Kameda,Takarazuka-shi, all of Japan Field of Search ..260/ 309.5

References Cited UNITED STATES PATENTS 3,134,663 5/1964 Kl'Oll..260/309.5

FOREIGN PATENTS OR APPLlCATlONS 1,018,261 10/1957 Germany ...260/309.51,032,258 6/1958 Germany. ...260/309.5 629,779 10/1963 Belgium..260/309.5

OTHER PUBLICATIONS Lombardino et al., J. Med. Chem. Vol. 7, pages 97-I0] (1964). RSLJS Paul at al., Chem. Abst. Vol. 53, columns 2272922730(1959). QDLAS] Primary Examiner-Natalie Trousof Attomey-Stevens, Davis,Miller & Mosher [57] ABSTRACT3-(3',5'-Dihalogenophenyl)imidazolidine-2,4-dione derivatives, which maybe substituted with alkyl or phenyl in the 1 and/or 5 positions, are newcompounds, and have high microbicidal activity on various fungi andbacteria and are non-toxic to plants and mammals. Those compounds areprepared by the intramolecular cyclization of a corresponding ureaderivatives.

9 Claims, No Drawings 3-( 3 ,5 '-DIHALOGENOPHENYL)IMIDAZOLH)lNE-2,4-DIONE DERIVATIVES The present invention is concerned with 3-(3,5'-dihalogenophenyl) imidazolidine-2,4-dione derivatives havingmicrobicidal activities which are represe n te d bythe fo qnula,

wherein X is a halogen atom; R is hydrogen atom, an alkyl group havingone to five carbon atoms, or a phenyl group; R, is hydrogen atom, analkyl group having one to six carbon atoms which may be substituted withmethylmercapto group, phenyl group or benzyl group; and R is hydrogenatom, an alkyl group having one to six carbon atoms, or phenyl group; aprocess for preparing said derivatives; and novel fungicides containingsaid derivatives as active ingredients.

The novel compounds of the present invention which represented by theformula (I) can be easily prepared by intramolecularly cyclizing at anelevated temperature in the presence of an acid catalyst a ureaderivative represented by the formula,

wherein X, R,, R and R are as defined previously, and R is carboxylgroup, a lower alkoxycarbonyl group or cyano group.

Some 3-phenylimidazolidine-2,4-dione derivatives have been known ascompounds, among which there are those having herbicidal activities.However, all the present compounds are novel and have no detrimentalaction on plants at all, and it has been found that they have strong andextremely broad microbicidal activities which have never been expectedfrom other homologous compounds.

That is, there has been attained such a surprising novel knowledge thatonly in the case where halogen atoms have simultaneously beensubstituted in the 3- and 5-positions of the benzene ring attached tothe nitrogen atom of the imide moiety of a3-phenylimidazolidine-2,4-dione derivative, does the said derivativedisplay physiological activities entirely different from those of otherhomologous compounds, i.e., strong and broad fungicidal activities, andit has no detrimental action on plants at all.

The present compounds have prominent effects on such a wide scope offungi as Blast, Sheath blight, Helmihthosporium leaf spot and bacteriumas Bacterial leaf blight of rice. plants/and fungi as Sclerotinia rot,Powdery mildew, brown rot, Black spot, Alternaria leaf spot, gray mold,Blossom blight, Canker and Damping-off of vegetables and fruits ofagricultural and horticultural crops. Moreover, they not only have nodetrimental action on useful plants and fishes but also are entirelyfree from toxicity to mammals, unlike mercury or arsenic preparations,and hence are quite excellent as plant pathogenic microbe-controllingagents. In addition, the present compounds can effectively controlAspergillus niger which propagates in industrial products, etc.

An object of the present invention is to provide the abovementionednovel compounds.

Another object is to provide novel and useful microbicides.

Other objects will become apparent from the following description.

The present compound of the formula (I) can be easily prepared byheating the urea derivative of the formula (ll) in the presence of anacid catalyst such as hydrochloric acid, sulfuric acid and phosphoricacid, under reflux.

The reaction is almost completed within 3 hrs.

Typical examples of the compounds represented by the formula (l), whichare provided in accordance with the present nvention, ar a l ows:

2 3-( 3,5'-dichlorophenyl) imidazolidine-2,4-dione 3-( 3 ,5-dichlorophenyl )-5-methylimidazolidine-Z,4-dione 3-( 3 ',5'-dichlorophenyl )-5-isobutylimidazolidine-2,4-dione 3-( 3' ,5'-dichl0rophenyl )-5-hexylimidazolidine-2,4-dione 3-( 3 ,5-dichlorophenyl )-5-( 2-methylmercapto) ethylimidazolidine-2,4-dione 3-(3 ,5 '-dichlorophenyl )-5-benzylimidazolidine2,4-dione 3-(3,5-dichlorophenyl)- l -methylimidazolidine-2,4-dione 3-( 3,5-dichlorophenyl)-1-ethylimidazolidine-2,4-dione3-(3,5'-dichloropheny1)-l-isopropylimidazolidine-2,4-

dione 3-( 3 ,5 '-dichlorophenyl )-5 ,5-dimethylimidazolidine-2,4-

dione 3-( 3 ,5'-dichlorophenyl I -methyl-5,5-

dimethylimidazolidine-Z,4-dione 3-( 3',5 '-dichlorophenyl)-5-phenylimidazolidine-Z,4-dione 3-( 3 ,5 '-dichlorophenyl)- l-phenylimidazolidine-2,4-dione 3-( 3 ,5 -dibromophenyl )-imidazolidine-2,4-dione 3-( 3 ,5 '-dibromophenyl )-5-phenylimidazolidine-2,4-dio ne Inactual application as microbicides, the present compounds may be used ina pure form without incorporation of other ingredient, or, for easierapplication as microbicides,

' concentrates, granules, etc. Further, they may be used in admixturewith other chemicals such as, for example, Blasticidin S, Kasugamycin,Polyoxin, pentachlorobenzaldoxime, 'yl ,2,3 ,4,5,-hexachlorocyclohexane, N-( 3 ,5 -dichlorophenyl) maleimide,N-(3',5'-dichlorophenyl) succinimide, N-( 3,5- dichlorophenyl)itaconimide, 0,0-diethyl-S-benzyl phosphorothioate, O-ethyl-S,S-diphenylphosphorodithioate, O-butyl-S-benzyl-S-ethyl phosphorodithioate,O-ethyl-O-phenyl-O-(2,4,5-trichlorophenyl) phosphate, 0,0-dimethyl-O-(3-methyl-4-nitrophenyl) phosphorothioate, S-[ l,2-bis(ethoxycarbonyl)ethyl]-0,0-dimethyl phosphorodithioate, 0,0-dimethyl-S-(N-methylcarbamoyl methyl) phosphorodithioate,0,0-diethyl-O-(2-isopropyl-6-methyl-4-pyrimidinyl) thiophosphate,3,4-dimethylphenyl-N-rnethyl carbamate, zinc ethylenebisdithiocarbamate, N-trichloromethylthio-4- cyclohexene-1,2-dicarboximide,N-( l l ,2,2- tetrachloroethylthio )-4-cyclohexenel ,Z-dicarboximide,iron methylarsonate, l-( N-n-butylcarbamoyl )-2-methoxycarbonylaminobenzimidazole, etc. and, in every case, no controlling effects ofindividual chemicals are decreased. Accordingly, simultaneous control oftwo or more diseases and/or injurious insects is possible andsynergistic effects due to the mixing can be expected. In addition, thepresent compounds may be used in admixture with such agriculturalchemicals as herbicides, nematocides, mitecides, etc. or withfertilizers.

The present invention is illustrated in further detail below. It is,however, needless to say that the kinds of starting materials for thepresent compounds, the reaction conditions and the kinds and mixingproportions of additives to be incorporated into the present compoundsare not limited only to those set forth in the examples but are variableover a wide scope.

EXAMPLE 1 Preparation process:

a. Standard operational process.

0.1 mole of a urea derivative represented by the formula (ll) and 50 g.of 20 percent hydrochloric acid are mixed in a ml. four-necked flask,and the mixture is stirred with reflux for 2 hours. After completion ofthe reaction, a deposited crystal is recovered by filtration, is washedseveral times with water and is dried, whereby a desired 3-(3,5-dihalogenophenyl) imidazolidine-2,4-dione derivative is obtained in afavorable yield. If necessary, the derivative may be recrystallized fromethanol-water to obtain the desired product higher in purity.

b. The urea derivatives represented by the formula (II), which are usedin the present process, can be easily obtained by reacting, according toordinary procedure, 3,5- dihalogenophenyl isocyanates with correspondingamines. Typical examples of the phenyl isocyanates and amines are asshown below, but it is needless to say that the scope o f thg presentinvention is not limited thereby. Names of compounds:

1. Phenyl isocyanates:

3,5ditluorophenyl isocyanate 3,5-dichlorophenyl isocyanate3,5-dibromophenyl isocyanate 3,5-diiodophenyl isocyanate 2. Amines:

glycine N-methylglycine N-ethylglycine N-n-propylglycine N-methylglycineethyl ester glycine ethyl ester glycine-n-butyl ester aminoacetonitrileN-n-propylaminoacetonitrile a-aminopropionitrile alanine N-methylalanineN-methyl-a-methylalanine N-ethylalanine alanine ethyl esterN-benzylalanine ethyl ester N-benzyl-a-methylalanine ethyl esterN-phenyl-a-methylphenylalanine glutamic acid ethyl N-methylglutamatea-aminopropionitrile a-methylaminopropionitrile a-amino-n-butyric acida-benzylamino-n-butyric acid ethyl a-amino-n-butyratea-amino-n-butyronitrile a-isopropylamino-n-butyronitrilea-amino-isobutyric acid ethyl a-aminoisobutyrate ethyla-methylamino-a-ethyl-isobutyrate a-amino-isobutyronitrile valineN-phenylvaline oz-methylvaline N-methyl-a-methylnorvaline norvalineethyl ester a-amino-n-valeronitrile oz-isopropylamino-n-valeronitrileleucine N-n-propyl isoleucine leucine ethyl estera-amino-isocapronitrile isoleucine N-ethyl isoleucine isoleucine ethylester norleucine N-benzyl-a-methylnorleucine norleucine ethyl estera-amino-n-capronitrile oz-methylamino-n-capronitrile a-amino-n-caprylicacid ethyl a-isopropylamino-n-caprylate a-ethylamino-n-capronitrilephenylalanine phenylalanine ethyl ester N-phenyl phenylalaninea-amino-B-phenylpropionitrile c. The present compounds were synthesizedaccording to the above-mentioned preparation process to obtain theresults set forth in Table l.

EXAMPLE 2 Preparation of microbicidal compositions:

The compounds employed in this example are represented by the numbers ofthe compounds set forth in Table l of the preceding Example 1.

A. Dusts:

3 parts of each of the compounds (1) and l2) and 97 parts of y were o hy pulv r zed ns m ififisz s hsma h tain 2 kinds of dusts containing,individually, 3 percent of active ingredient. In application, the dustswere dusted as they were.

B. Dusts:

4 parts of each of the compounds (4) and l3) and 96 parts of talc werethoroughly pulverized and mixed together to obtain 2 kinds of dustscontaining, individually, 4 percent of active ingredient. lnapplication, the dusts were dusted as they were.

C. Wettable powders:

50 parts ofeach ofthe compounds (6) and 14), 5 parts ofa wetting agentof alkylbenzenesulfonate and 45 parts of diatomaceous earth werethoroughly pulverized and mixed together to obtain 2 kinds of wettablepowders containing, individually, 50 percent of active ingredient. Inapplication, the wettable powders were diluted with water and thesolutions were sprayed.

D. Emulsifiable concentrates:

10 parts of each of the compounds (10) and (15), parts of dimethylsulfoxide and 10 parts of an emulsifier of polyoxyethylene phenylphenolether were mixed together to obtain 2 kinds of emulsifiable concentratescontaining, individually, 10 percent of active ingredient. Inapplication, the emulsifiable concentrates were diluted with water andthe emulsions were sprayed.

E. Granules:

5 parts ofeach of the compounds (7) and (16), 93.5 parts of clay and 1.5parts of a binder of polyvinyl alcohol were thoroughly pulverized andmixed together, kneaded with water and then granulated and dried toobtain 2 kinds of granules containing, individually, 5 percent of activeingredient.

F. Composite dusts:

2 parts of each of the compounds l) and (14), 1.5 ofO-nbutyl-S-ethyl-S-benzylphosphorodithiolate, 0.1 part of Kasugamycinand 96.4 parts of clay were thoroughly pulverized and mixed together toobtain two kinds of dusts containing, individually, 3.6 percent ofactive ingredients. ln application, the dusts were dusted as they were.

G. Composite dusts:

2 parts of each of the compounds (4) and 18), 1.5 parts ofO-n-butyl-S-ethyl-S-benzylphosphorodithiolate, 2 parts of 0,0-dimethyl-O-(3-methyl-4-nitrophenyl) phosphorothioate, 1.5 parts of3,4-dimethylphenyl-N-methylcarbamate and 93 parts of clay werethoroughly pulverized and mixed together to obtain 2 kinds of dustscontaining, individually, 7 percent of active ingredients. Inapplication, the dusts were dusted as they were.

EXAMPLE 3 Typical test results of the present compounds are shown belowwith reference to test examples in order to substantiate the facts thatthe characteristic physiological activities of the present compounds areobserved only in the case where a 3,5- dihalogenophenyl group has beensubstituted in the nitrogen atom of the imide body, and no suchactivities are observed at all in the case where other phenyl group hasbeen substituted therein: and that the present compounds have strong andbroad microbicidal efiects which have never been seen in the smisna ns sqss- TEST EXAMPLE 1 Effects of controlling rice blast:

Rice plants (variety: WASEASAl-llflwhich had been cultivated to the 3leaves stage in flower pots of 9 cm. in diameter, were sprayed with 7ml. per pot of each of aqueous solutions of test compounds in the formof emulsifiable concentrates. After one day, the plants were sprayed toinoculate with a spore suspension of rice blast fungus (Pyriculariaoryzae). 5 days thereafter, the number of diseased spots generated wascounted to investigate the fungicidal effects of individual testcompounds. This test was carried out on different days to obtain theresults as set forth in Tables 2 and 3. As is clear from Tables 2 and 3,the present compounds showed markedly excellent results as compared withthe control known compounds. i WW FWFM,

TABLE 5 7 TABLE 4 -(un\inucd M H m w 1 Hi OHC CH0 0H0 C8 N r r l 1 B B 00 0 w w w 0 0 U H c .r H H H H Nllno N C 0 0 0 0 A 0 0 C 0 N N d M W M nB U U 5 U 2 a 3 w 0 W 0 U 2 B, 2 2 k a i w m H m m HIT... N n 0 0 0 0 &C 0 0 0 H C N m I. H W. l r. o u 0 0 2 n w u m u m M m U 0 0 U 3 5 M H Hi m I w u 1 I u m M o w N YO H m H 0 u (w 0 H Ha o nw U IIIIIUIIU H w "Mr I.- 1| b H C 0 N 0" C" N n /N\ m 0 m l m N C H n n m r m M Controlknown com ound.

TABLE 5 Test. compound of cm., the plants were sprayed with 10 ml. perpot of nm 6 n mm b e h ma hm 2 n. h i c a w m H \hr m mm H m m m H a LAm P .w E n m M .lo M H A h n X m W m u s H H s c H 0 S m n s C 3 ME aMn H N c m o T w p m w w u "m m; Mn 0" C 0 0 0 C 0 m m h w N N t 0 G n nn w:| 2 G k f m1 0 e N t m t w t B m u m c .w m C W h 01 E U N C 5 0 5 O5 5 6 6 7 7 3 o u 1. 3 0 0 0 z 2 2 m m a C C a H /H\ H C 2 0 z 0 a H H HH H H W J ml mi n C 0 0 0 0 0 0H0 CH 0 0 H N N N N 1 u m m m m m m eachof aqueous emulsions of test compounds in the form of emulsifiableconcentrates. After 3 hours, mycelial pieces disc of 5 cm. in diameterof rice sheath blight fungus (Pellicularia rasakii) were inoculated ontothe leaf sheaths of individual stems. 5 days thereafter, the size ofeach diseased spot (re gion) on the leaf sheaths was measured, and thedegree of damage of each stem was calculated to investigate thefungicidal effects of individual test compounds. This test was carriedout on different days to obtain the results as set forth in Tables 6 and7. As is clear from Tables 6 and 7, the resent compounds showed markedlyexcellent controlling effects as compared with the control knowncompounds.

The degree of damage was calculated according to the following equation:I

Zllndex of infectionX Number of stems) Number of investigated stems X 3X 100 wherein the index of infection is:

index 0 No spot was observed on the leaf sheaths. l Spot-like shadeswere observed on the leaf sheaths.

l0 (trace) TABLE 6 Active ingredient concentra- Degree of Test compoundtlon (p.p.m.) damage I C-NH 0-011; 01 0 200 3 8 Cl I] C-NH l C-CHCHe 01ll O 200 5.8 CI

/PJNH N\ /CH: (In fi1- HCH: CH\

O 200 7 6 Cl 1 NH x l C-CHCHzCHrBCH: C] [i /CNH l C-CHCHz- C1 il 0 a 200100 o /C-NH I C-CH; 0

0 200 2 01 73 l /C-NH' O 200 89 1 01 H l /C-NH l Q-CHCH: I!

TABLE 8 (nnlinuml TABLE 7 -(onlinucd Active ingredient, concentrn-Degreoof Active ingredient concentra- Degree of n (p p m damage Testcompound Test compound TUZ' (commercially available !ungicide)Non-treatment- When pumpkin plants (variety: HEIAN KOGIKU), which 40 hadbeen cultivated in flower pots of 12 cm. in diameter,

24 TABLE 9 Continued TABLE 8 Cniinued Active ingredient conccntra-Degree of damage Test compound tlon (p.p.m.)

Test compound Non-treatment-....

TABLE 9 5 Km 4 m s "m: il i D n m 0 0 U c m m N m 0 mm m m e t .r... w nC 0 N 5 0 2 3 w 4 new 8 e r w n D eb 0 mmm no .10. t

H c z H a 0 0 CH0 m 0 p m c w C C T TEST EXAMPLE 5 Effects ofcontrolling cucumber damping-off )2 Flower pots of 9 cm. in diameterwere individually packed with farm soil. On the surface of said soil wasuniformly spread each 10 m1. of pathogenic soil in which had beencultured and propagated cucumber damping-off fungus (Pelliculariafilamentosa). To this soil was applied 15 ml. per pot of each of Numberof non-infected seedlings in each treated area Number of seedlingsgerminated in area non-treated and unlnoculated lated according to thefollowing equation:

Percentage of stand TABLE 10 TABLE 10-C0ntinued Active ingredientconcentrnr Percentage Test compound tion (prp.m.) of standPentachloronitrobenzenc (commercially available fungicide)- 500 all. s)N tin-treatment (fungus inoculated) o Non-treatment (uninoculated) 100TABLE 11 TABLE 1;

Active 10 Inhibiingredient fury eoneoncentra- Percentage vent ru- Tcstcompound tron (p.p.m.) of stand lion 'lest llllCl'OOl'gillllSlllsi.p.in.) 0 CH3 500 90. 1 01 H Rice blast fungus (Pyricularia 071/200) 4.t, M or m 20H CNCH 15 Rico bacterial leaf blight fungus (Xanlhmuunax"Tl/Cal). f) Cucumber seedling (lfllllplllgOll fungus (Iellicnlaria liln- N CH1 mentosa) 40 Peanut southern sclerotiuin rot fungus (Corticiumrolfsz" V 200 C-CH2 Strawberry gray mold fungus (Botrytis cincrea) t u tM ()1 li Rape sclerotinia rot fungus (Sclerotinia sclcrofiorum) t 4 0Pear black spot fungus (Allcmaria kikuthfana). Apple Alternaria leafspot fungus (Altt'rnaria mall)... 200 20 Grape ripe rot fungus(Glomzrclla cingulata) e 200 Peach brown rot fungus (Sclerotinia cinema)n X 500 92. Cl ll I What IS claimed [Si l 1. A 3-(3',5'-dihalogenophenyl)imidazolidine-2,4-dione l CCOH3 derivativerepresented by the formula, 01 ll 0 (13H: 0 X ll l CNR| 0 500 100 Br llN C-NH I CCRz N l X ll l l O R: C-CH2 Br '5 wherein X is halogen; R ishydrogen, alkyl of one to five carbon atoms or phenyl; R is hydrogen,alkyl of one to six carbon 0 500 0 atoms which may be substituted withmethylmercapto, phenyl H or benzyl; and R is hydrogen, alkyl of one tosix carbon atoms or phenyl. 2. A3-(3',5'-dihalogenophenyl)imidazolidine-2,4-dione 40 derivativerepresented by the formula, ll 01 i I tLNH 0 CH 500 7. 5 Q Cl H CN-CH Clu 1 l 0 Cl N CHs 3. A 3-(3,5'-dihalogen0phenyl)imidazolidine-2,4-dionell derivative represented by the formula,

0 Br g 0 500 6. 6 NH u C-NH i 5 5 C0 II t- I ll IL 0 (fi-(f-CIIJ 0 0 CH4. A 3-(3',5'-dihalogenophenyl)imidazolidine-2,4-dione derivativerepresented by the formula,

Pentaehloronitrobenzene (commercially 500 86.4 0 available fungicide).G1 Non-treatment (fungus inoculated) 0 Non-treatment (uninoculated) 100-N Control known compounds.

I C 0 HC H 01 ll TEST EXAMPLE 6 5. A3-(3',5-dihalogenophenyl)imidazolidine-2,4-dione derivative representedby the formula,

Anti-microbial spectrum: C l

According to the agar medium dilution method, the fungus JJNCH;growth-inhibiting effects of 3-( 3',5'-dichlorophenyl)-2,4-imidazolidine dione on various plant pathogenic fungi and bacteria wereinvestigated to obtain the results as set forth in 0-011 1 II Table 12.0

6. A 3-(3,5'-dihalogenophenyl)imidazolidine-2,4-dione 8. A3-(3,5'-dihalogenophenyl)imidazolidine-2,4-dione derivative representedby the formula, derivative represented by the formula,

C1 ii Cl A N C H I 5 I a 1(1s Q i C- H: C- CHa 01 01 g l 0 9. A 3-(3,5-dihalogenophenyl)imidazolidine-2,4-dione 7. A 3-(3,5'-dihalogenophenyl)imidazolidine-2,4-dione derivative represented bythe formula, derivative represented by the formula, 0

2. A 3-(3'',5''-dihalogenophenyl)imidazolidine-2,4-dione derivativerepresented by the formula,
 3. A3-(3'',5''-dihalogenophenyl)imidazolidine-2,4-dione derivativerepresented by the formula,
 4. A3-(3'',5''-dihalogenophenyl)imidazolidine-2,4-dione derivativerepresented by the formula,
 5. A3-(3'',5''-dihalogenophenyl)imidazolidine-2,4-dione derivativerepresented by the formula,
 6. A3-(3'',5''-dihalogenophenyl)imidazolidine-2,4-dione derivativerepresented by the formula,
 7. A3-(3'',5''-dihalogenophenyl)imidazolidine-2,4-dione derivativerepresented by the formula,
 8. A3-(3'',5''-dihalogenophenyl)imidazolidine-2,4-dione derivativerepresented by the formula,
 9. A3-(3'',5''-dihalogenophenyl)imidazolidine-2,4-dione derivativerepresented by the formula,